Problem 81 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(g\) is a function with domain \([-4,10],\) what is the domain of the function \(2 g(x-1) ?\)

Step-by-Step Solution

Answer

The domain of the function \(2g(x-1)\) is \([-3,11]\).

Step 1: Understand the given domain

The original function is given as having a domain of \([-4,10]\). This means that the function \(g(x)\) is defined for all \(x\) values between -4 and 10, inclusive.

Step 2: Consider the transformation inside the function

The new function is \(2g(x-1)\). Notice that \(g(x-1)\) involves a horizontal shift. Specifically, \(g(x-1)\) shifts the graph of \(g(x)\) to the right by 1 unit.

Step 3: Determine the new bounds

Since the shift is right by 1 unit, every \(x\) value in the original domain will increase by 1. Therefore, the lower bound of the domain will be \(-4 + 1 = -3\), and the upper bound will be \(10 + 1 = 11\).

Step 4: State the new domain

After considering the horizontal shift, the new domain for the function \(2g(x-1)\) is \([-3,11]\).

Key Concepts

horizontal shift

In mathematics, a horizontal shift occurs when a function's graph is moved left or right along the x-axis. This is done by modifying the function's input variable, typically denoted by \(x\). If you add a number inside the function, such as \g(x + c)\, you shift the graph to the left by \(c\) units. Conversely, if you subtract a number, like in \g(x - c)\, the graph shifts to the right by \(c\) units.

For example, consider the function \(g(x-1)\) inside our given problem. This transformation shifts the graph of \g(x)\ to the right by 1 unit. So every point on the graph of \g(x)\ will be displaced one unit to the right. This transformation affects the domain of the function directly, which we'll cover in more detail in the next sections.

function transformations

Function transformations involve changing the position or shape of a graph on a coordinate plane. There are several types of function transformations, including horizontal shifts, vertical shifts, reflections, stretches, and compressions.

In our given problem, we are dealing with two transformations:

• The horizontal shift, by replacing \(x\) with \(x-1\) inside the function \(g(x)\).
• A vertical stretch, by multiplying the result of \(g(x-1)\) by 2.

Whenever you stretch the graph vertically by a factor, such as 2, each point on the graph is moved twice as far from the x-axis. Note that these transformations do not alter the domain, but they significantly change the graph's appearance.

domain and range

The domain and range of a function are crucial concepts in understanding function behavior. The domain refers to all possible input values (x-values), while the range refers to all possible output values (y-values). In the context of our problem, we started with the domain \([-4, 10]\)\ for the function \(g(x)\).

When we apply the horizontal shift, \(g(x-1)\), each x-value in the original domain is increased by 1. This adjustment shifts the entire domain right by 1 unit. So the new domain is determined as follows:

• Lower bound: \(-4 + 1 = -3\)
• Upper bound: \(10 + 1 = 11\)

The new domain of our transformed function \(2g(x-1)\) is \([-3, 11]\). This informs us that the function \(2g(x-1)\) is defined for x-values between -3 and 11, inclusive.

Understanding the relationship between domain and transformations helps in mapping out how inputs change under various function manipulations.